More results on the signed double Roman k-domination in graphs

Aequationes Mathematicae, vol. 99, pp. 1903–1921

Abstract

Abstract Let $$k\ge 1$$ k ≥ 1 be an integer, and let G be a finite and simple graph with vertex set V(G). A signed double Roman k-dominating function (SDRkDF) on a graph G is defined in [Signed double Roman k-domination in graphs, Australas. J. Combin. 72 (2018), 82–105] as a function $$f :V(G) \rightarrow \{-1,1,2,3\}$$ f : V ( G ) → { - 1 , 1 , 2 , 3 } satisfying the conditions that $$\sum _{x\in N[v]}f(x)\ge k$$ ∑ x ∈ N [ v ] f ( x ) ≥ k for each vertex $$v\in V(G)$$ v ∈ V ( G ) , where N[v] is the closed neighborhood of v, every vertex u for which $$f(u)=-1$$ f ( u ) = - 1 is adjacent to at least one vertex v for which $$f(v)=3$$ f ( v ) = 3 or adjacent to two vertices x and y with $$f(x)=f(y)=2$$ f ( x ) = f ( y ) = 2 , and every vertex u with $$f(u)=1$$ f ( u ) = 1 is adjacent to vertex v with $$f(v)\ge 2$$ f ( v ) ≥ 2 . The weight of an SDRkDF f is $$\textrm{w}(f) = \sum _{v\in V(G)}f(v)$$ w ( f ) = ∑ v ∈ V ( G ) f ( v ) . The signed double Roman k-domination number $$\gamma _{\textrm{sdR}}^k(G)$$ γ sdR k ( G ) of G is the minimum weight among all SDRkDF on G. In this paper we continue the study of the signed double Roman k-domination number of graphs, and we present new bounds on $$\gamma _{\textrm{sdR}}^k(G)$$

Authors 2

1. Michael A. Henning corresponding

   University of Johannesburg

   Affiliation as printed

   Department of Mathematics and Applied Mathematics, University of Johannesburg, Auckland Park, 2006, Johannesburg, South Africa

2. Lutz Volkmann Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University, 52056, Aachen, Germany